本文關(guān)鍵詞： 分?jǐn)?shù)階Laplace算子 MEMS方程 邊緣場(chǎng) 上下解 解的存在性　出處：《華東師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：在本文中,首先我們將用延拓的方法重新給出分?jǐn)?shù)階Laplace算子的定義,接下來會(huì)敘述并給出分?jǐn)?shù)階方程上下解方法的證明,最后將討論帶有邊緣場(chǎng)的分?jǐn)?shù)階MEMS方程解的存在性.在第三章中,我們主要討論1/2階Laplace算子,首先將N維空間內(nèi)的區(qū)域Ω延拓為N+1維空間內(nèi)的柱形區(qū)域D,然后通過跡算子定義一個(gè)新的空間∧(Ω)并定義空間內(nèi)函數(shù)的調(diào)和延拓.通過一系列等價(jià)性證明,我們可將帶有Dirichlet邊界條件的非局部問題轉(zhuǎn)化為帶有Neumann邊界條件的局部問題,由此得到1/2階方程弱解的定義.在證明了分?jǐn)?shù)階方程解的存在性上下解方法之后,我們將用上下解方法重點(diǎn)討論帶有邊緣場(chǎng)的分?jǐn)?shù)階MEMS方程解的存在性,在討論λ足夠小時(shí),我們會(huì)轉(zhuǎn)而討論一個(gè)近似方程的解的存在性.
[Abstract]:In this paper, we first give the definition of fractional Laplace operator by extension method, then we describe and prove the method of upper and lower solutions of fractional order equation. Finally, we discuss the existence of solutions for fractional MEMS equations with edge fields. In chapter 3, we mainly discuss the Laplace operators of order 1/2. Firstly, the domain 惟 in N-dimensional space is extended to the cylindrical region D in N-dimensional space. Then we define a new space A (惟) by trace operator and define the harmonic extension of functions in the space, and prove it by a series of equivalence. We can transform a nonlocal problem with Dirichlet boundary condition into a local problem with Neumann boundary condition. The definition of the weak solution of the 1/2 order equation is obtained. After the existence of the upper and lower solutions of the fractional order equation is proved. We will focus on the existence of solutions for fractional MEMS equations with edge fields by using the upper and lower solution method. When we discuss the sufficient number of 位, we will turn to the existence of solutions for an approximate equation.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)博士學(xué)位論文 前1條

1 席莉靜;含非局部算子的橢圓邊值問題及相關(guān)問題解的存在性研究[D];蘇州大學(xué);2014年

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本文編號(hào)：1472268